The paging problem is that of deciding which pages to keep in a memory of k pages in order to minimize the number of page faults. We develop the marking algorithm, a randomized on-line algorithm for the paging problem. We prove that its expected cost on any sequence of requests is within a factor of 2Hk of optimum. (Where Hk is the kth harmonic number, which is roughly ln k.) The best such factor that can be achieved is Hk. This is in contrast to deterministic algorithms, which cannot be guaranteed to be within a factor smaller than k of optimum. An alternative to comparing an on-line algorithm with the optimum off-line algorithm is the idea of comparing it to several other on-line algorithms. We have obtained results along these lines for the paging problem. Given a set of on-line algorithms and a set of appropriate constants, we describe a way of constructing another on-line algorithm whose performance is within the appropriate constant factor of each algorithm in the set.